The Australia Telescope Compact Array (ATCA) has been used for high angular resolution imaging of 71 southern class I methanol maser sources quasi-simultaneously at 36 and 44GHz. The data reveal a high level of morphological and kinematical complexity, and allow us to demonstrate associations, at arcsecond precision, of the class I maser emission with outflows, expanding HII regions, dark clouds, shocks traced by the 4.5-micron emission and 8.0-micron filaments. More than 700 maser component features were found at each of the two methanol transitions, but with only 23 per cent recognisable at both transitions; the morphology of class I emission is much better revealed by our survey of both transitions, compared with either one alone. We found that the number of masers falls exponentially with the projected linear distance from the associated class II 6.7-GHz methanol maser. This distribution has a scale of 263+/-15-milliparsec, irrespective of the transition. The class I masers associated with OH masers were found to have a tendency to be more spread out, both spatially and in the velocity domain. This is consistent with the expectation that such sources are more evolved. Apart from a small number of high-velocity components (which are largely blue-shifted and predominantly seen at 36GHz), the velocity distribution was found to be Gaussian, peaking near the systemic velocity of the region, which had been estimated as the middle of the velocity interval of the associated class II methanol maser at 6.7GHz. The mean indicated a small, but significant blue shift asymmetry of -0.57km/s (uncertainties are 0.06 and 0.07km/s for the 36- and 44-GHz masers, respectively) with respect to the 6.7GHz masers. The standard deviation of the velocity distribution was found to be 3.65+/-0.05 and 3.32+/-0.07km/s for the 36- and 44GHz masers, respectively. We also suggest a refined rest frequency value of 36169.238+/-0.011MHz for the 4_-1_-3_0_E methanol transition.